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Factor 360, a highly composite number with many divisors.
Key values: 360 = 2^3 x 3^2 x 5 · 24 divisors · phi(360) = 96
Check if 7919 is prime (it is -- the 1000th prime number).
Key values: 7919 is prime · 2 divisors · phi(7919) = 7918
Compute the GCD and LCM of 84 and 120.
Key values: 84 = 2^2 x 3 x 7 · 120 = 2^3 x 3 x 5 · GCD = 12, LCM = 840
Enter an integer greater than 1 to see its prime factorization, factor tree, and divisors.
Given the prime factorization , the number of divisors is:
Each divisor is formed by choosing an exponent from 0 to for each prime factor. The product of all these choices gives the total count.
Example: , so divisors.
This uses the geometric series formula for each prime power component.
Example: .
A perfect number satisfies — the sum of its proper divisors equals the number itself.
Euclid-Euler theorem: Every even perfect number has the form where is a Mersenne prime. Whether odd perfect numbers exist is one of the oldest unsolved problems in mathematics.
Find the prime factorization , then compute . For example, has divisors.
Use the formula for each prime power in the factorization. For , .
A perfect number is a positive integer that equals the sum of its proper divisors, which means . The first three perfect numbers are 6 (= 1+2+3), 28 (= 1+2+4+7+14), and 496. By the Euclid-Euler theorem, every even perfect number has the form where is a Mersenne prime.
Highly composite numbers have more divisors than any smaller positive integer. They tend to be products of the smallest primes with descending exponents, like (6 divisors), (12 divisors), and (16 divisors).
The divisor function is multiplicative, meaning whenever . This follows from the fact that each divisor of uniquely corresponds to a pair of divisors, one from and one from , when and share no common prime factor.